The Horseshoe prior is a continuous shrinkage prior, and hence block structure recovery is not straight-forward. In Bayesian fusion estimation with Laplace shrinkage prior or with \(t\) -shrinkage prior, one may discretize the samples obtained via MCMC using a suitable threshold. [1]} recommended using the \(1/2n\) -th quantile of the corresponding prior for discretization of the scaled samples. In Section , we proposed a threshold based on the posterior contraction rate. However, that would also require an idea of the true block size, which may not be available always, or may be difficult to ascertain beforehand. We provide a practical approach for discretization of the samples. Note that in a sparse normal sequence problem, the posterior mean of the mean parameter \(\theta _i\)  is given by \(\kappa _i y_i\) , where \(\kappa _i\)  is the shrinkage weight. These shrinkage weights mimic the posterior selection probability of the means, and hence thresholding the weights to 0.5 provide excellent selection performance, which is justified numerically [2]} and later theoretically [3]}. Motivated by the same, we propose to use the following thresholding rule for selection of the blocks for our proposed method: \(\theta _{j_1}\)  and \(\theta _{j_2}\)  are equal if \(|\hat{\theta }_{j_1} - \hat{\theta }_{j_2}| < 0.5|y_{j_1} - y_{j_2}|\) , where \(\theta _{j_1}, \theta _{j_2}\)  are the posterior means of \(\theta _{j_1}\)  and \(\theta _{j_2}\) respectively, for \(1 \le j_1 \ne j_2 \le n.\)  We thus estimate the block structure indicator \(s_{j_1j_2} = 1\!\mathrm {l}\lbrace \theta _{j_1} = \theta _{j_2}\rbrace \)  by \(\hat{s}_{j_1j_2} = 1\!\mathrm {l}\lbrace |\hat{\theta }_{j_1} - \hat{\theta }_{j_2}| < 0.5|y_{j_1} - y_{j_2}|\rbrace \) . To evaluate the performance of the structure recovery method using the proposed thresholding approach, we define the metrics \(W\)  and \(B\) , where \(W\)  is the average within-block variation defined as \(W := \mathrm {mean}_{\lbrace s_{j_1j_2} \ne 0\rbrace }|\hat{\theta }_{j_1} - \hat{\theta }_{j_2}|\) , and \(B\)  is the between-block separation defined as \(B := \min _{\lbrace s_{j_1j_2} = 0\rbrace }|\hat{\theta }_{j_1} - \hat{\theta }_{j_2}|.\)
